Let $G$ be a finite group, $p$ a prime, $P$ a Sylow $p$-subgroup of $G$ and $d$ a power of $p$ such that $1<d<\left|P\right|$. Let $G_p^{\ast }$ denote the unique smallest normal subgroup of $G$ for which the corresponding factor group is abelian of exponent dividing $p-1$. Let ${𝔉}_1$, ${𝔉}_2$, ${𝔉}_3$ be classes of all $p$-groups, $p$-nilpotent groups and $p$-supersolvable groups, respectively, $G^{𝔉}$ be the $𝔉$-residual of $G$. Let $X\in \{{(G_p^{\ast })}^{{𝔉}_1},G^{{𝔉}_2},G^{{𝔉}_3}\}$. A subgroup $H$ of a finite group $G$ is said to have $\mathrm{\Pi }$-property in $G$, if for any $G$-chief factor $L/K$, $|G/K:N_{G/K}((H\cap L)K/K)|$ is a $\pi ((H\cap L)K/K)$-number. A normal subgroup $E$ of $G$ is said to be $p$-hypercyclically embedded in $G$ if every $p$-$G$-chief factor of $E$ is cyclic, where $p$ is a fixed prime. In this paper, we prove that $E$ is $p$-hypercyclically embedded in $G$ if and only if for some $p$-subgroups $H$ of $E$, $H\cap X$ have $\mathrm{\Pi }$-property in $G$.

让$G$是一个有限群，$p$一个素数，$磷$西洛$p$-子群$G$和$d$一种力量$p$这样$1<d<\left|磷\right|$. 让$G_p^{*}$表示唯一的最小正规子群$G$对应的因子组是指数除法的阿贝尔$p-1$. 让${𝔉}_1$,${𝔉}_2$,${𝔉}_3$成为所有人的班级$p$-团体，$p$- 幂零群和$p$- 超可解组，分别，$G^{𝔉}$成为$𝔉$-剩余的$G$. 让$X\in \{{(G_p^{*})}^{{𝔉}_1},G^{{𝔉}_2},G^{{𝔉}_3}\}$. 一个子群$H$有限群的$G$据说有$\mathrm{\Pi }$- 财产$G$, 如果有的话$G$-主要因素$\mathrm{大号}/ķ$,$|G/ķ：{ñ}_{G/ķ}((H\cap \mathrm{大号})ķ/ķ)|$是一个$\pi ((H\cap \mathrm{大号})ķ/ķ)$-数字。正态亚组$乙$的$G$据说是$p$- 超循环嵌入$G$如果每个$p$-$G$- 主要因素$乙$是循环的，其中$p$是一个固定素数。在本文中，我们证明$乙$是$p$- 超循环嵌入$G$当且仅当对于某些人$p$-亚群$H$的$乙$,$H\cap X$有$\mathrm{\Pi }$- 财产$G$.